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5 comments
One of many problems with this (besides the fact that we can just numerically check that π≠17-8√3) is stated here in the top comment.
On a more fundamental level we know that π cannot be represented in terms of algebraic numbers (17-8√3 is an example of an algebraic number) because π was proven to be transcendental by Lindemann in the 1800s. (Transcendental numbers are by definition not algebraic.)
√3 is exact. In math we define numbers by what they do, not by how many digits of them we can write down for instance, we have defined √3 to be a unique positive real number such that (√3)2 =3, we didn't define it by saying it's close to 1.73205. However, knowing the first few digits of an irrational number can be useful, it's just important not to conflate the approximation with what the number actually is.
One reason why we define things this way is that it doesn't really make sense to work with real numbers just by looking at their decimal expansions; their structure is far too rich to be captured by merely looking at where they fall on the number line.
For instance take [char]#937[/char] =0.56714... It doesn't look very special but if I instead tell a mathematician that it's a unique real number that satisfies ΩeΩ =1 he/she gets more clarity.
tl;dr we don't define irrational numbers through their approximations/decimal expansions
Confused?
So were we! You can find all of this, and more, on Fundies Say the Darndest Things!
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